Math Problem Statement

5(3-2i)+2i(4+6i)=

Solution

To simplify the expression 5(32i)+2i(4+6i)5(3 - 2i) + 2i(4 + 6i), we’ll expand each term and then combine like terms.

Step 1: Expand 5(32i)5(3 - 2i)

5(32i)=5352i=1510i5(3 - 2i) = 5 \cdot 3 - 5 \cdot 2i = 15 - 10i

Step 2: Expand 2i(4+6i)2i(4 + 6i)

2i(4+6i)=2i4+2i6i=8i+12i22i(4 + 6i) = 2i \cdot 4 + 2i \cdot 6i = 8i + 12i^2 Since i2=1i^2 = -1, replace 12i212i^2 with 12(1)=1212 \cdot (-1) = -12: 2i(4+6i)=8i122i(4 + 6i) = 8i - 12

Step 3: Combine Terms

Now, add the results from Step 1 and Step 2: 1510i+8i1215 - 10i + 8i - 12

Combine like terms (real parts and imaginary parts):

  • Real part: 1512=315 - 12 = 3
  • Imaginary part: 10i+8i=2i-10i + 8i = -2i

Final Answer

32i3 - 2i

Would you like a deeper explanation or have any further questions?

Here are five related questions to consider:

  1. How would the answer change if ii were replaced with a different imaginary unit?
  2. What is the geometric interpretation of the complex number 32i3 - 2i on the complex plane?
  3. How can we factor expressions involving complex numbers?
  4. How do we multiply complex numbers using polar form?
  5. What other mathematical properties are unique to complex numbers?

Tip: Always remember that i2=1i^2 = -1; this substitution is essential in simplifying expressions with complex numbers.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Algebraic Expansion
Imaginary Unit

Formulas

Distributive Property a(b + c) = ab + ac
i^2 = -1

Theorems

Properties of Complex Numbers
Arithmetic of Complex Numbers

Suitable Grade Level

Grades 10-12